Asymmetric imaging through engineered Janus particle obscurants using a Monte Carlo approach for highly asymmetric scattering media

The advancement of imaging systems has significantly ameliorated various technologies, including Intelligence Surveillance Reconnaissance Systems and Guidance Systems, by enhancing target detection, recognition, identification, positioning, and tracking capabilities. These systems can be countered by deploying obscurants like smoke, dust, or fog to hinder visibility and communication. However, these counter-systems affect the visibility of both sides of the cloud. In this sense, this manuscript introduces a new concept of a smoke cloud composed of engineered Janus particles to conceal the target image on one side while providing clear vision from the other. The proposed method exploits the unique scattering properties of Janus particles, which selectively interact with photons from different directions to open up the possibility of asymmetric imaging. This approach employs a model that combines a genetic algorithm with Discrete Dipole Approximation to optimize the Janus particles' geometrical parameters for the desired scattering properties. Moreover, we propose a Monte Carlo-based approach to calculate the image formed as photons pass through the cloud, considering highly asymmetric particles, such as Janus particles. The effectiveness of the cloud in disguising a target is evaluated by calculating the Probability of Detection (PD) and the Probability of Identification (PID) based on the constructed image. The optimized Janus particles can produce a cloud where it is possible to identify a target more than 50% of the time from one side (PID > 50%) while the target is not detected more than 50% of the time from the other side (PD < 50%). The results demonstrate that the Janus particle-engineered smoke enables asymmetric imaging with simultaneous concealment from one side and clear visualization from the other. This research opens intriguing possibilities for modern obscurant design and imaging systems through highly asymmetric and inhomogeneous particles besides target detection and identification capabilities in challenging environments.


Modeling the particle using discrete dipole approximation
To calculate the scattering properties of the particles inside the cloud, we resort to the well-known Discrete Dipole Approximation (DDA) [1], [2], [3].In this approach, the scatter is discretized into N lattice points with size d x , and the jth element presents polarizability tensor   = (3 0   3 (  −   )/(  + 2  )), where   ,   ,  0 are lattice, surrounding medium and vacuum permittivity, respectively, as shown in Figure S2.Using the framework of the DDA, the dipole moment of the jth nano-antenna cell   is given by where   (  ) is the incident electric field at the jth lattice position   ,   is the Green-tensor from the jth to the nth element, given by, When equation 2.2 is expressed for the N lattice points, it becomes a linear system solved using numerical approaches to compute   , such as the generalized minimal residual method (GMRS), or the quasi-minimal residual method (QMRM).When the particles are inside the cloud, the scattered photon could collide with the particle from any incident direction (  ,   ), therefore, to compute for all the possible situations, we compute   (  ,   ) for all possible incidences, given by Note that E0 is chosen so the incident wave has power equally distributed between p-and s-polarization.After solving the linear system, the extinction cross-section ( }. (1.4) where   is the total power dissipated by the particle,  0 is the background wave impedance, ω is the angular frequency and   is the intensity of the incident wave (  = 1/2 0 ).Note that the extinction coefficient of a cloud is given by   (  ,   ) =   (  ,   )   , where   is the particle density.The radiation intensity (  ,   , , ) for a given incident wave is (  ,   , , ) =  We can think that the scatter behaves as an antenna, where it redirects an incoming photon in the direction   ,   to a new direction ,  with efficiency  =   (  ,   )/  (  ,   )part of the photons are absorbed and dissipated by the particle.Moreover, the directivity gives information on the new photon direction since it behaves as a probability distribution function.In this sense, we can calculate the probability of a photon being forward-or backward-scattered, given its incoming directions, as seen in Figure S2.
Assuming a photon arriving from side A (B), the total forward ( ()  ) and backward ( ()  ) scatterings are calculated as (1.9) . (1.10) Finally, the total forward ( ()  ) and backward ( ()  ) scattering from a photon arriving from an external illumination source with angle α is given by, . (1.11) Up to this point, we have calculated the properties of a single particle.However, the single-particle properties are related to the

Reciprocity Theorem
To prove the reciprocity, we start by expanding the propagating electric field at the point  in terms of electromagnetic multipoles as follows, where m and l are integers corresponding to the multipoles,    and    are the regular electric and magnetic multipoles, respectively, and    and    are the electric and magnetic multipoles moments, respectively.In the manuscript, the particles are excited by a plane wave propagating with an elevation and azimuthal angles (  ,  ).The electric field of the plane wave can be written by substituting the multipole moments as follows, , (2. 4)   where,    (cos   ) is the associated Legendre polynomial.Note in (4), that when the wave arrives from opposites sides,    ( −   ) = (−1) +    (  ) and    ( −   ) = −(−1) +    (  ).
To prove the reciprocity theorem, we need to prove that the cloud extinction   (  ) when   = 0° is equal to when   = 180°.Note that the particle can be oriented in any direction, holding this as a general proof.The light scattered by the particle when the light arrives from   = 0° (  + ) and   = 180° (  − ) are given by, where   ,± and   ,± are the electric and magnetic multipole moments of the scattered wave, respectively.The most common approach to calculate    and    is by means of using the T-matrix.Using this technique, the    and    can be related to the moments of the incident wave by a matrix multiplication operation, as follows, (2. 7) where  ± and  ± are the truncated multipole moments for scattered field ( = 1. .),   and   are the truncated multipole moments of the incident field, and AA, AB, BA and BB are the matrices that's related the incident field moments with the scattered field moments.Considering a TM incidence (the same procedure can be done for TE), equations ( 2 As a property of the T-matrix,  ,l ,`= (−1) +  ,` `, ,  ,l ,`= (−1) +  ,` `, and  ,l ,`= −(−1) +  ,` `, [4].In this sense, (2.12) can be written as )    * ]} =   + (2.13) Since   ± =   ±   , where N is the particle density (1/m 3 ) and   is the cloud length, eq.(2.13) proves that the ballistic attenuation of the cloud when the photon arrives from opposite sides are the same.

Sun Position
We have considered the same number of photons for all radiation sources on the Monte Carlo approach.However, some of these powers have different values than others, and here we show the relation between them.Figure S2 shows that the cloud is illuminated by the external source of power (I 0 ) and by a single pixel on side A (P A ). Considering L A and L B as the distance between the target on side A and the observer on side B to the cloud, respectively, and also NA as the numerical aperture of the camera used by the observer, the maximum aperture angle (αmax) radiated by PA that can be scattered to B side can be calculated as, Note that we consider   in our Monte Carlo Approach, and all photons are generated to be inside this region.From this, we can calculate the total power radiated by I0 to the cloud (note that in our approach, I0 is normalized by the total number of pixels   where W0 is the power density (W/m 2 ) of I0, and   = (2     ) 2 is the area illumination.The primary illumination source also illuminates PA, which absolves Aabso of the photons and scatterers all photons isotropically.However, we only considered in the Monte Carlo the photons inside going to the illumination area.In this sense, the P A can be calculated using the relation  (  /(  +   )<0.5),  0 /  becomes higher than  0 /  .This represents that the I0 sends more noise photons to side B (since the power of  0 /  is higher) then to side A. In this sense, positioning the cloud closer to the target on side A would also help increase the contrast ratio, which is the primary goal of the manuscript.

Figure
Figure S1-Discretized Particles for Direct Dipole Approximation method.
obtained by exchanging LA and LB on the equations.FigureS3shows  0 /  (blue) and  0 /  (red) as a function of the relative distance position of the cloud between side A and B (  /(  +   )) and considering   = 0.5 and NA=0.33 as inside the main manuscript.When the cloud is at the center, the impact of the photons generated by the external illumination is the same on both sides of the cloud, and a factor of 18 needs to be applied.When the cloud moves close to side A

Figure S2 -
Figure S2 -Cloud being illuminated by the external source of power (I0) and by a single pixel on side A (PA).LA and LB are the distance between the target on side A and the observer on side B to the cloud, respectively

Figure S6 -
Figure S6 -Target image used for PD/PID calculations.
and the total radiated power (  (  ,   )) can be calculated using, With   (  ,   ), we can calculate the scattering cross-section of the particle (  (  ,   )) and the directivity ((  ,   , , )), E0 is the amplitude of the electric field, TM and TE denote transversal magnetic and transversal electric, and where